The asymptotic decay of the magnitude of the Fourier transform of a function appears always to be determined by its continuity properties as follows, with examples given in Fig. 1:
- Continuous integral, -20 dB / decade;
- Continuous function, -40 dB / decade;
- Continuous 1st derivative; -60 dB / decade;
How can this be expressed mathematically in a rigorous way, and proved?
Figure 1. dB magnitude of the Fourier transform of a function as function of the base-10 logarithm of frequency for (from slowest to fastest decay) a rectangular pulse (turquoise), a single lobe of a cosine (purple), and Hann function (blue).